ECE 6560 Multirate Signal Processing Fourier Transform Review
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1 ECE 656 Mutirate Siga Proessig Fourier Trasfor Revie Dr. Bradey J. Bazui Wester Mihiga Uiversity Coege of Egieerig ad Appied Siees Departet of Eetria ad Coputer Egieerig 93 W. Mihiga Ave. Kaaazoo MI,
2 Fourier Trasfor Operatios Cotiuous Tie ad Frequey Fourier Trasfor Disrete Tie ad Cotiuous Frequey Fourier Trasfor Disrete Tie ad Disrete Frequey Fourier Trasfor Iverse Trasfor Proofs Iverse Idea LPF DTCF Fourier Trasfor Eape ECE Bradey J. Bazui
3 Referees [] Sait K. Mitra, Digita Siga Proessig, A Coputer-Based Approah, 3rd Ed., MGra-Hi, 6. ISB: [] Proais ad Maoais, Digita Siga Proessig: Priipes, Agoriths, ad Appiatios, 4th Ed., Pretie Ha, 7. ISB: [3] A. V. Oppehei ad R. W. Shafer, Digita Siga Proessig, Pretie-Ha, 975. ISB: ECE Bradey J. Bazui 3
4 Cotiuous-Tie/Cotiuous-Frequey Forard Trasfors Iverse Trasfors a a a t ep t f t ep i f t t a ept t f epi f t ECE Bradey J. Bazui 4 dt dt d df
5 Disrete-Tie/Cotiuous-Frequey CTCF Trasfor Sapig operatio Derivatio: Variabe Chage a a a t ep t e ep ECE Bradey J. Bazui 5 a a dt t t t t t t t ep t t t t ep t a a t ep t tep t t dt dt
6 Frequey Repiatio of Disrete-Tie/CF Trasfor e ep e ep ep e Therefore there is spetra repiatio i frequey. arg e For ora oputatios, e defie the priipe vaue of the phase for ECE Bradey J. Bazui 6 arg e, for
7 Disrete-Tie/Cotiuous-Frequey Forard Trasfor Iverse Trasfor e ep e ep d ECE Bradey J. Bazui 7
8 ECE Bradey J. Bazui 8 Iverse DTCF Trasfor The iverse is prove as foos: (are the d bouds uique?) d e ep d ep ep ˆ d ep ep ˆ d ep ep ˆ ep ep ˆ si os si os ˆ si ˆ si ˆ,, si ˆ
9 ECE Bradey J. Bazui 9 Cotiuous- to Disrete-Frequey Trasfor Derivatio Substitutig for speifi disrete frequeies t t ep s f f t ep W ep t t f s t a t t ep f f s The Disrete Fourier Trasfor otatio
10 ECE Bradey J. Bazui The Iverse DTDF (DFT) Taig the iverse of the forard trasfor ep W ep ep ˆ ep ˆ,, ep ˆ ˆ
11 Equivaee of Frequey Rage The DTCF Frequey boud either [ to π] or [- π to + π] otig that This aso ipies a boud of [ to f s ] or [-f s / to + f s /] The DTDF t The frequey steps are = ( to -) otig that f The frequey steps are t ECE Bradey J. Bazui f s fs f f f s
12 Disrete ad Cotiuous Spetra Frequey Ais Equivaee Pot ad visuaizatio of a yquist bad-iited o pass siga ad the adaet spetra iages/repias Based o the trasforatio used the siga oud be portrayed ith ay oe of the ais represetatios. ECE Bradey J. Bazui
13 Trasfors or Rea Iputs A rea siga iput resuts i a ougate syetri output. The poer spetra desity (agitude) is a irror iage aroud zero. The phase is ati-syetri, a egative irror about zero. f t ep i f t dt f t os f t dt i t si f t dt Re I f Re f f I f ep os si ECE Bradey J. Bazui 3 Re Re Re I I Iportat property: oy ½ of the resut is eeded to defie the opete resut! I
14 Rea-yquist Bad Liited sapig for Siusoida IF Sigas yquist Criteria for a IF etered at f s /4 ote that the egative frequey opoet oud be etered at either -f s /4 or 3f s /4 based o the frequey ais dra. The trasitio bad fods or aiases ito the passbad spae; therefore a reaisti passbad is ess tha yquist (f s /). ECE Bradey J. Bazui 4
15 Cope-yquist Bad Liited sapig for Siusoida IF Sigas yquist Criteria for a IF etered at f s / ote that there is o required spetra syetry for a ope iput! The trasitio bad fods or aiases ito the passbad spae; therefore a reaisti passbad is ess tha yquist (f s ). ECE Bradey J. Bazui 5
16 Disrete Frequey Trasfors Bi aiget o uit steps for the ope iput DFT. Aterate haf bi aiget (Odd Frequey) fs Digita IF FFT Aaysis Ces OFFT Aaysis Ces Whih oe aes ore sese for eistig RF sigas? ECE Bradey J. Bazui 6
17 ECE Bradey J. Bazui 7 Itroduig the OFFT The Odd-Frequey Fourier Trasfor Syetry for rea () i e i e i i e e i i e e o e i i e
18 Properties The DFT has uerous properties that ao ay tris to be perfored. Frequey shiftig Syetry (rea data, syetri data, ati-syetri data, et.) Hardare ipeetatios (dua rea proessig, haf-egth ope proessig, et.) DST ad DCT a be perfored as DFTs ECE Bradey J. Bazui 8
19 Perforig Spetra Iversio (spetra represetatio ad ath) db oest freq. highest freq. Iverted Spetru IF Passbad fs/ ope iig Desired Spetru -8 db -fs/ -fs/4 fs/4 fs/ y i e, for y i e y i e y ECE Bradey J. Bazui 9
20 Spetra Shift up by Fs/4 Miig fro basebad (=) to Fs/4 (=/4) Mae a rea siga ope! Reated to Weaver Moduatio for SSB y i e, for 4 y i 4 e Eapes: DC etered siga upshiftig or IF siga doshiftig y y i e i ECE Bradey J. Bazui
21 Ipeetatio The ath equatios are ie, but e usuay at effiiet hardare ipeetatios. Addressig ad storage are heap Mutipies ad adds are epesive Ca e aipuate the ath to perfor feer oputatios? Eiiate redudat (or ougate) oputatios! ECE Bradey J. Bazui
22 FFT Syetry for Rea Data W W * Y i W y Y i W Y Y * y i W y i W y Y Y * () () ECE Bradey J. Bazui
23 Usig Syetry The Dua Rea Data FFT Z W iy Z iy (3) Y iagz reay iagz Z iagy reaz Z reaz Y iagz rea Z rea rea Z Z iag iag Z Z (4) Etrat to rea trasfors fro oe ope trasfor ECE Bradey J. Bazui 3
24 OFFT Syetry for Rea Data * W W W W W W () Aterate foruatio for sipifiatio (/ poit trasfor) W W W W W W ECE Bradey J. Bazui 4 W W W W i ()
25 Usig Syetry Aterate Ipeetatio of OFFT Aterate Foruatio Cotiued W W * W i W i (3) W / Deay / Deay FFT Co ECE Bradey J. Bazui 5
26 Disrete Cosie Trasfor (oputatio usig FFT) DCT Soutio os i W i * 4 4 W i W W * i W W ECE Bradey J. Bazui 6
27 DST Soutio Disrete Sie Trasfor (oputatio usig FFT) si i i W i * 4 4 i W i W W * i W W ECE Bradey J. Bazui 7
28 Idea Frequey Doai LPF (DTCF-FT) ECE Bradey J. Bazui 8 ret,, d ep d ep ep ep ep si si si Proais ad Maoais, Digita Siga Proessig: Priipes, Agoriths, ad Appiatios, 4th Ed., Pretie Ha, 7. ISB:
29 Idea Frequey Doai LPF Probes The sape doai futio is defied for a sape tie fro ifiity to +ifiity. A pratia fiter aot ast that og so et s ut-it-off soehere. ie at ad +. h H iit si ret Gibbs Pheoeo si ep Proais ad Maoais, Digita Siga Proessig: Priipes, Agoriths, ad Appiatios, 4th Ed., Pretie Ha, 7. ISB: ECE Bradey J. Bazui 9
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